Reference no: EM132731366
Question: Describing the properties of your cubic surface and the elliptic curves on this surface, including appropriate computer-generated images. Please make your title interesting and include an abstract at the beginning of the paper. Submit as pdfs and electronic files in CANVAS (i.e. pdf+ source MSword, Latex files, (as text) , ppt, etc. Include your computer codes if you use them for calculations.).
The points colored in purple are to be turned in as a draft on Dec 9th for the review and input. They will be graded as midterm work and revised to be included in your final paper.
Below is the structure of the paper. The poster should summarize your work in a form that can be presented at a conference next semester (see the attached template).
Start your paper with points 1 and 2 below, and then follow the other requirements (you can change the order).
Your number b = 11, 13, 17, 19 , 23 and your p = 11 .
y2 = x3 - 3x + b
1. Title, name, abstract - (write this part at the end highlighting your results).
2. Give basic definitions of P2(x,y,z,), an algebraic variety in P2 , singular points, a dimension a variety.
3. Give the homogenous equation f of your curve V in P2 (x,y,z). What is the space classifying all degree 3 curves in P2 ? Ax3 +B y3 + C z3 + D x2y + E y 2z + F xz2 +Gxyz = 0 (how coefficients. P6)
4. What is the dimension of this classifying space? Show the calculations.
5. Consider y2 = x3 - 3x + b for b= 11, 13, 17, 19 , 23. Give definitions of Ux, Uy, Uz, Graph your curves for , x=1, y=1, z=1 (use any graphing software).
6. Find singular points of V in P2 or prove there are none. Find the inflection points on your curve or prove there are none.
7. Pick a non-singular point and write the equation of a tangent space. What dimension is your variety at non-singular points?
8. What is the dimension of a family of cubic curves of the form y2 = x3 - ax + b? Explain.
9. Consider y2 = x3 - 3x + b for b= 11, 13, 17, 19 , 23. Calculate the genus g for each curve and the j - invariant for C. (The genus formula for a smooth curve on a plane is g= (d-1)(d-2)/2 , where d is a degree of the polynomial defining the curve).
10. Graph the curves as a plane curve in R2 over Z11 . Include all points where 0< x < p and -p < y < p. Make a table showing all solutions. Give total points on each curve for all b's.
11. On each curve pick a pint P on C. Calculate P+P=2P, P+2P = 3P, .... mP=0 (use the formulas in the handout). What is the order of m of P (or the order of the cyclic group {P, 2P, 3P, .... mP=0} ? Can you find P with the largest m on your curve? Make tables.
12. Any connection between m and b?
13. Choose a curve and use (m-1)P from the previous point where m is the largest you could find. Pick any point Q on C and calculate the code using (m-1)P for the message to be send. (m-1)P+Q
14. Anything interesting about the situation?
15. Bibliography - cite all sources you have used, including our textbook and a calc book.
Attachment:- CURVEsrypto.rar